Why Is Nicomachus' Identity Special?

Abstract

Nicomachus' identity $\sum_{k=1}^{n} k^3 = \left(\sum_{k=1}^{n} k\right)^2$ is familiar, but the structure behind it is less often emphasized. We begin with Nicomachus' original pattern, in which successive cubes are expressed as successive blocks of consecutive odd integers, and show how triangular numbers naturally enter the resulting sum-of-cubes identity. We then reverse the usual triangular-number argument and start from a general difference of squares representing $n^p$ as a sum of $n$ consecutive odd integers. This construction is possible for every $p\geq 2$, but only for $p=3$ do the two squares correspond to consecutive triangular numbers, which explains why Nicomachus' odd-number blocks fit together without gaps or repetitions. Finally, using the leading terms of the Faulhaber polynomials, we show that, apart from the trivial case, Nicomachus' identity is the unique relation of the form $S_s(N)=S_1(N)^r$ for positive integer exponents.